Refraction

When light enters a different medium, its wavelength changes according to the refractive index of the new medium. The relationship between the wavelength in air ( ) and the wavelength in a medium ( ) is given by: Where:
- is the wavelength of light in air,
- is the refractive index of water. Substitute the known values: Thus, the wavelength of the light in water is .

We use Snell's law to solve this problem, which relates the angle of incidence and refraction between two media: Where:
- and are the refractive indices of the two media,
- is the angle of incidence in the first medium,
- is the angle of refraction in the second medium.
Step 1: Conditions for Grazing Refraction At the glass
-liquid interface, the ray is refracted out grazing the face AD, meaning the angle of refraction at the glass
-liquid interface must be .
Step 2: Using Snell’s Law at the Glass
-Liquid Interface
- The refractive index of glass is ,
- The refractive index of the liquid is ,
- The angle of refraction . Using Snell's Law at the interface:
Step 3: Calculating the Angle Now, calculate the angle of incidence : Thus, the angle of incidence .

When a glass slab is placed over an object, the apparent shift in position is related to the thickness of the slab and its refractive index. The formula for the apparent shift is given by: where:
- is the thickness of the slab,
- is the refractive index of the material. Given:
- ,
- . Substituting the values: Thus, the shift in the letters printed on the paper is 2 cm.

Geiger-Marsden Scattering Experiment: The Geiger-Marsden experiment, conducted in 1909 by Hans Geiger and Ernest Marsden under the guidance of Ernest Rutherford, involved the scattering of alpha particles by a thin gold foil. The primary objective of the experiment was to investigate the structure of the atom by studying how alpha particles interact with matter. In the experiment: - A beam of alpha particles (helium nuclei) was directed at a thin gold foil. - A detector surrounded the foil to detect the scattered alpha particles at various angles. - The alpha particles were expected to pass through the gold foil with little deflection if the atom was composed of uniform matter. However, what they observed was surprising: - While most of the alpha particles passed through the foil with only slight deflections, some alpha particles were deflected by large angles, and some even bounced back at angles close to 180°. This observation led to the conclusion that the atom is not made of uniformly distributed matter, but rather has a small, dense, positively charged core at its center. This core was later identified as the nucleus, which contains most of the atom’s mass. Graph Showing Variation of Number of Scattered Particles with Scattering Angle: The graph showing the number of scattered particles as a function of scattering angle typically has the number of particles (N) on the y-axis and the scattering angle ( ) on the x-axis. The graph is generally bell-shaped, with most of the alpha particles being scattered at small angles (close to 0°), while fewer particles are scattered at large angles. The number of particles scattered at very large angles (e.g., 180°) is minimal.

The sharp increase in the number of particles scattered at small angles and the small number of particles scattered at large angles led to a significant breakthrough in understanding atomic structure. Discovery of the Nucleus: The results of the Geiger-Marsden experiment led Rutherford to propose that the atom has a small, dense, positively charged nucleus at its center. The fact that a small fraction of alpha particles were scattered at large angles suggested that there was a very small, dense, positively charged center within the atom (the nucleus) that repelled the positively charged alpha particles. This was in stark contrast to the previously accepted plum pudding model, where the positive charge was assumed to be spread out evenly throughout the atom. Thus, the Geiger-Marsden experiment, combined with Rutherford’s analysis, directly contributed to the discovery of the atomic nucleus and the development of the nuclear model of the atom.

Let’s consider the ray diagram for a parallel sided glass slab: 1. A ray of light is incident on the glass slab at an angle . 2. The ray is refracted inside the glass at an angle . 3. The ray then emerges out of the slab at the same angle (because the glass slab is parallel-sided).

From the diagram, we see that after the light emerges from the slab, it undergoes a lateral shift due to the thickness of the slab. Derivation of Lateral Shift: Let the lateral shift be , which is the horizontal displacement of the emergent ray from the original path. The light undergoes refraction twice: 1. First, when it enters the slab at angle , it refracts to angle . 2. Second, when it exits the slab, it refracts back to the same angle . Let the thickness of the slab be , and the distance traveled inside the slab be (since the ray is traveling at angle ). The lateral shift is given by the following formula: where: - is the angle of incidence, - is the angle of refraction, - is the thickness of the slab. Condition for Minimum Shift: The lateral shift will be minimum when the angle of incidence is equal to the angle of refraction , i.e., when . This would make the ray pass straight through the slab without any lateral shift. Thus, the condition for the minimum lateral shift is when the incident angle is equal to the refracted angle ( ).

Concept: Refractive index of a medium: Also: Where:

(speed of light in vacuum)
= speed of light in medium

Step 1: Calculate refractive index.
Step 2: Find speed of light in liquid. Final Answer: